Bismut type formulae for differential forms

TL;DR

This paper derives Bismut type formulae for the heat semigroup acting on differential forms, expressing derivatives in terms of Brownian motion expectations without involving curvature derivatives.

Contribution

It provides explicit Bismut type formulae for the derivatives of the heat semigroup on differential forms, avoiding curvature derivatives.

Findings

Formulas for $dP_t $, $d^*P_t $, and $ riangle P_t $ derived

Expressions involve Brownian motion expectations with random translations

Curvature derivatives are not needed in the formulae

Abstract

Formulae are given for $dP_t \phi$, $d^*P_t\phi$ and $\Delta P_t\phi$ for $P_t$ the heat semigroup acting on a q-form $\phi$. The formulae are Brownian motion expectations of $\phi$ composed with random translations determined by Weitzenbock curvarure terms. Derivatives of the curvature are not involved.